Amaton

Not really. 16 does has two distinct square roots, as does every other real number (besides zero, which has only itself). Nothing will change that. However, notation is not flexible and we denote specific things in specific ways according to convention. The radical [math]\sqrt{n}[/math] almost always denotes only the principal value. The following statements are true according to the typical convention. [math]\sqrt{16}=4[/math] [math]\sqrt{16}\ne \pm 4[/math] [math]\pm\sqrt{16}=\pm 4[/math] The latter of the three is how one should express both the positive and negative values.

Ah. So it basically comes down to whether or not the ion has unpaired valence electrons? Or is there more to it?

That's a pretty interesting question. [math]2\pi[/math] radians forms a complete revolution in planar angles, and [math]4\pi[/math] steradians do the same in 3-dimensions. How many steradians comprise the interior solid angles of a cube? Or a dodecahedron? And is there a general formula for regular polyhedrons as there is for the planar analog?

I appreciate your convenience. Thanks for that. I'm just befuddled by the explanation behind the method, but I must admit the concept and the work is very cool. It's likely just my shortcomings. Maybe the wording is a little misleading? First of all, the visible spectrum is just the visible spectrum. As far as I know, electromagnetic radiation does not have any sort of octave equivalency as sound does. Saying that the data was dropped 40 octaves tells me nothing -- What is this supposed to mean precisely? In terms of wavelength or frequency? Then, "dropped each light frequency precisely forty octaves into the human auditory range" -- disregarding whatever 40 octaves of light means, how can electromagnetic radiation become sound? I suspect a specialized conversion process was used. Did you mean that the frequency was lowered enough so that once it was converted to sound data it would be audible to the human ear? Here, it's a matter of causality in the statements, semantics in a way.

I thought it was a cool notion also, but I'm still not comfortable with this idea so I'll just come out with it... Okay. It would make sense if your friend were using data from acoustic spectrometry (representation via sound waves) but according to his website, he's using light. Then I thought, well maybe your friend is using some nontrivial method by means of specialized software, but according to this phrasing you seem to claim he's done a direct transposition -- of light... into sound. How is this possible? Since when can one lower the frequencies of EM radiation down to sound? Please tell me I'm not as dull as I suspect.

Ahem... Wonders if it may be a form of EM. Wonders if it might be a chemical change. Now claims that it must be EM. Then claims it is the motion of matter. Any sincere user, whether they're inquiring or asserting, can at least hold a coherent viewpoint. I honestly wonder if this is a troll.

Pssh, you call that a strong? We tea extremists prefer tea solvent diluted with some water, thank you very much.

I may just be dull in the mind, but I have to ask... Am I the only one who found something odd with this? ---EDIT--- Yeah, nevermind.

An interesting topic -- though irrelevant, it seems. I think daniton meant "perfect square" as in any [math]n\in\mathbb{Z}[/math] such that [math]\sqrt{n}\in\mathbb{Z}[/math] i.e. 1, 4, 9, 16, 25, ... Anyway, I don't understand the initial post. Is OP attempting to show Fermat's infamous unknown proof?

Simply "blanking" a spherical volume within a solid iron ball in normal conditions wouldn't change much given the sturdy outer structure. Earth's solid core, on the other hand, is still pretty hot and under extreme pressure. That immense pressure might collapse the iron-nickel mass into it from all directions, thus shrinking or completely eliminating the blank. EDIT: This is still a speculatory answer, though, given by analogy. I'd wait to see what an expert might say on it since physical changes at such extreme conditions are less likely to follow intuition. Also, dense iron-nickel solid isn't exactly as "malleable" in shape as one might think, due in part to whatever kind of molecular structure it might hold.

Replicators would be quite the thing, but a publicly-accessible distribution seems less likely than the technology itself. I would think governments will have a rather strict regulation over that. They'll have to. One product of analogy that we have today is far from that, but it might have that impact in the long run. 3D Printers. Just look what they're doing now, and they're on the rise.

If the singularity event turns out as its proponents believe, one might think of significant consequences much more significant than those affecting the business sector. Seeing as it's a rather flexible prediction, the outcomes and impacts on anything can be approached in different ways. I've barely gotten any depth into the idea, and indeed many think the event will have a huge impact on the economy, increasing the rate of exponential growth in the global economy. Personally, I don't see how one could draw that out so directly. Thoughts?

I agree with ACG52. I don't know about the forum's policy on this, but it is preferred that one provides the information here rather than reference his/her own source. The article was rather short and could be summarized, even written out entirely with due formatting, right here. (EDIT: Considering how you phrased the original post, it certainly does seem like self-promotion. Hope mods take quick action) Nonetheless, the question seems sincere, and I find it interesting. There are, in fact, such scenarios if one looks at it a certain way. Abstract statements certainly exist whose validities are impossible to determine with a given set of axioms. The most famous of these may be Cantor's Continuum Hypothesis. It's not a paradox, per say, but it can be neither proven nor falsified within the standard framework of mathematics, which is ZFC+Choice. See this: List of statements undecidable in ZFC. In light of theoretical physics, the mathematics therein is very difficult and complex. Almost everyone suspects that a Theory of Everything will be formulated in the language of mathematics, just as all the major and matured theories of physics have been. (how such a unifying theory can work without being mathematical in nature, I don't know) Before Einstein's renowned work, there were observations that simply could not be explained by previous theory. Then General Relativity emerged with development in differential geometry, and it is very accurate and consistent with what it concerns. Though the subject is miles ahead of me, I highly doubt it can be anything without due application of differential geometry. You can't mow the lawn with a sponge. Likewise, any unifying theory, not necessarily a ToE, may have to be approached differently in regards to mathematics. Noncommutative geometry is a branch of mathematics that concerns the geometry of noncommutative operations. This mathematical formulation will certainly work differently than any mathematics where (for sake of simplicity) multiplication is commutative, and thus it may be possible that such a theory can only be described consistently a certain way. See Noncommutative standard model.

Apparently there have been previous instances of this. So I'm curious -- in those prior escapes, what was the conclusion as to how the strains got out? Oh, and Monsanto's public image will only worsen now. Go figure.

Thanks. That certainly could be. With a quick search, I encountered a certain answer, a rather funny one, because it is from Yahoo!Answers. However, the contributor seemed knowledgeable in chemistry as per his other answers and established ranking (but this is a personal first for trusting Y!A). The question page is here. I'm surprised I have yet to find anything that explains either or their distinction from an authoritative organization. I took greater notice to the latter excerpt, "The electron beam generates, among other things, a positively charged molecule known as a molecular ion, which results from the removal of one electron from the molecule." It seems one can think of it as where the ion is derived from (?), i.e. molecular ions derive from molecular structures and thus molecular ions are a subset of polyatomic ions.

What exactly is a molecular ion? I ask because there appears to be nothing obvious that distinguishes a molecular ion from a polyatomic ion. I would be satisfied with personal inference but that merits nothing in regards to the correct and rigorous classification. I'm not sure if there's actually a subtle difference or if the terms are synonymous and thus interchangeable (the Wikipedia page seems to support the latter notion). If there is a distinction, I suspect it has to do with the type of bonding involved. Just looking for someone to shed light on these two terms. Thanks in advanced

Don't forget about Joe the track runner. Joe is doing a quick sprint. If Joe is 6 feet tall running at a sprinting steep of 75 degrees elevation, approximately how far will Joe's face travel if he trips and faceplants into the ground? (assume no gliding / air time)

Are you required to use Henry's Law in your answer? If so, that could majorly change your approach. Otherwise, I agree with EdEarl. I think the question is more endemic to chemical change than purely physical properties (i.e. how hemoglobin binding plays a role).

Elaborate what is meant by "business". A specific corporation focused on a particular technology? The broad market for that technology? Or the continuity of mega corporations in general?

Generally, the solubility of a gas in water decreases as temperature increases (assuming other conditions held constant). Assuming a constant temperature, solubility in water typically increases as pressure increases. So if you have a saturated solution of gas in water, an increase in container pressure will cause the solution to become supersaturated. Thus, the vapor pressure of the solution will increase and could be read by some device within the container. I'm not an expert, so is this valid reasoning?

Or depending on the context, e.g. in basic geometry, one would typically use the principal value, since the negative is intuitively meaningless for describing length, area, etc.

I see [math]a=bc[/math] as more of a matter of division than multiplication (might sound odd, but it's a matter of interpretation), assuming [math]b[/math] is a constant. Now, [math]b[/math] can be interpreted as a conversion factor, something which quantitatively relates [math]a[/math] and [math]c[/math]. So it really reflects the importance of direct proportionality between two variables, the fact that their quotient is always constant regardless of their size. It might be of interest to bring up [math]\pi[/math], since it is the constant equal to the quotient of a circle's circumference to its diameter. No matter how big or small the circle is, [math]C=\pi d[/math] is true. These kinds of expressions are also related to the also important equations of form [math]y=\frac{1}{2}cx^2[/math] via integration. Bringing up the circle example, one can derive [math]A=\pi r^2[/math] from the above identity for circumference. So now the question is... What is the fundamental significance of proportionality?

Who cares? It's a math problem, and the point is application (realistic or not). John bought 200 apples, then proceeded to eat 70 of them. Oh look, John now has 130 apples.

I don't believe that's necessary. Let's assume that there won't be any major obstacles to impede our long-term growth overall (or if there are, that we'll be able to recover quickly, relatively speaking for a few billion years). This should include natural disasters, mass war, etc. The exponential rate at which our technological capabilities grow should certainly make up in time. I'm not saying this on a strictly scientific basis, but we should reach a point where we can defend ourselves completely in the event of even a stellar system collision.

I see the Pythagorean means as a more mathematical-based grouping, rather than only historical, but that is sensible.